BSTJ 3: 4. October 1924: Some Contemporary Advances in Physics -V Electricity in Solids. (Darrow, Karl K.)

Some Contemporary Advances in Physics — V

By KARL K. DARROW

Electricity in Solids

IN considering such topics as the flow of electricity through solids
and the outflow of electricity across their boundaries, we have to
forego the assistance of the great system of laws, models, and word-
pictures which constitutes the contemporary theory of the structure
of the atom. This imposing and truly powerful theory, which now-
adays seems to bulk larger than all of the rest of physics, is after all
limited to certain restricted fields; it deals successfully with par-
ticular properties of isolated atoms, and also with certain qualities
of atoms which seem to be localized in their inner regions; but it avails
little or nothing in the study of the behavior of liquids and solids.
Much of the present-day theory of electrical conduction in solids is
based only on the very simplest assumptions as to the nature of the
atoms of which they are built, some would even remain valid under
the old-fashioned ideas of continuous electrical fluids; and profoundly
as we may believe that solids are built of atoms resembling Bohr's
famous model, it is highly doubtful whether that model has ever
helped to interpret a single one of the phenomena of conduction or
clone more than to provide a new language for old ideas.

We have first to make the distinction between the substances in
which atoms migrate along the path of the flowing current and ap-
parently carry the moving charge, and the substances in which the
atoms stand still while the current flows past them. It is universally
conceded that elements, and likewise the alloys of metals and a num-
ber of solid compounds, belong to the latter class; whatever it is that
carries the current flows through and past the substance, leaving it
at the end as it was at the beginning. Weber said in 1858, "In the
metals there are electrically-charged particles as well as atoms; some
of the former are freely mobile and others vibrate about the atoms;
they are the cause of the conduction of electricity and of heat, and of
magnetic phenomena as well." Considering that in Weber's day
electricity had never been observed apart from ponderable matter
and electrons were unknown, this is entitled to rank as a daring
anticipation.

Next we have to distinguish between conduction by metals and
conduction by non-metallic elements. Strictly we should begin by
defining a "metal"; but this task had better be left to the chemists,
as being really their affair; and they have found it no easy affair to

621

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SOME CONTEMPORARY ADVANCES IN PHYSICS— V 623

set up a definition by which every element can be confidently assigned
to one class or to the other. In fact there is a tendency to begin by
defining metallic conduction, and then define metals as the elements
which display it! The difficulty, as usual, is to make the definition
sharp enough to decide a few intermediate or transitional cases.
Anyone even slightly acquainted with chemistry or physics would
instantly recognize as metals the elements in the first column of
the Periodic Table, and those at the bottom of the table in all the
columns; and as non-metals, with the same ease, the elements in the
topmost row of the table and down the right-hand side. The first
element of every column after the first two is non-metallic, and the
non-metallic character advances farther and farther down the
columns as one proceeds across the Table from left to right. One
might say that the elements which are not metals occupy the north-
east sector of the Table, and the debatable ones cross in a diagonal
band from northwest to southeast. The elements which are gases
under the usual circumstances of temperature and pressure are
extreme instances of non-metals; but some of the definitely non-
metallic elements, and all of the debatable ones, are solid or liquid
under the usual conditions.

Very little could be said about the elements which under ordinary
conditions are gases, for very little is known about the manner in
which they conduct electricity when liquefied or frozen. Probably
the reason is that the experimental conditions would be unusually
difficult, and the substances probably very bad conductors; it is not
easy to imagine solid hydrogen moulded into a cylinder, drawn into
a wire, clamped or sealed between electrodes, or filled into a sheath
less conductive than the hydrogen itself. The difficulties may not
be insuperable; but they have not been generally overcome.

As for the solid elements which are definitely not metals, or which
belong to the debatable group, there is an abundance of data in
print, and yet not nearly so much as we need. In general their re-
sistances are tremendously greater than the resistances of metals;
"tremendously" for once is not an extravagant word, for the con-
ductivities of the elements are spread over a sweeping range of orders
of magnitude which few if any other qualities of theirs can rival.
The mass of the heaviest known atom differs from the mass of the
lightest only by a factor of 240; the densities of the solidified elements,
their compressibilities, their other mechanical and thermal proper-
ties range over not more than one or two, at the most three orders of
magnitude; even the energy required to extract the innermost electron
of an atom rises by a factor of only 10 5 in passing from the first to

624 BELL SYSTEM TECHNICAL JOURNAL

the last element of the series; but the conductivity of silver stands
to the conductivity of sulphur in the ratio 10 21 . The distance from
the sun to the nearest star is some 10 18 cm.; we see that a sheet of
sulphur a thousandth of an inch thick would offer more of an obstacle
to the passage of electricity than a cable of silver of the same diameter,
extending from the earth to Alpha Centauri. The variations of
conducting-power from element to element are thus as fantastically
great as the variations in scale from the world of common life to the
world of interstellar spaces. The conductivities of the metals, how-
ever, are confined within a narrow fraction of this range; it is between
the metals and the non-metals, and between one non-metallic element
and another, that the leaps are surprisingly great.

In general, too, the resistance of a non-metallic element decreases
as its temperature is raised; the curve of resistance versus tempera-
ture (I shall often call it characteristic, henceforward) slants down-
ward, the derivative and the temperature-coefficient of resistance
are negative. Near room-temperature this is the usual behavior,
but not always over the entire accessible range; of some elements it
is observed that the resistance declines less rapidly as the temperature
is raised, the curve is concave upward; eventually the decline ceases,
the resistance passes through a minimum value at a certain char-
acteristic temperature, and thereafter increases with the temperature
as the resistances of metals do. At least one element of the debatable
class (germanium) exhibits a characteristic curve that slants upward
instead of downward at room- temperatures ; but when the curve is
followed towards lower temperatures, it too is found to be concave
upward with a minimum of resistance below— 100° C. This suggests
that for all of the non-metals the resistance-temperature curve may
be a loop bulging downward, with a minimum at a certain temperature
that varies from element to element; on this generalization one of
the contemporary theories is founded.

These rules can be illustrated by mentioning briefly the behavior
of the non-metallic elements one by one. Beginning at the foot
of the procession of elements, we pass over hydrogen (no data), lithium
and beryllium (metals), and commence with boron. Boron has a very
high resistance at room temperature, which drops a hundredfold
when it is heated to 180° C. and ten-million-fold when it is raised to
a red heat. On carbon a tremendous amount of work has been done,
which unfortunately largely goes to show that the word "carbon"
usually signifies a framework of carbon atoms packed with occluded
gases, organic compounds, and impurities of divers kinds, which no
known mode of treatment avails to expel entirely, although almost

SOME CONTEMPORARY ADVANCES IN PHYSICS-V 625

anything which is done to the substance alters its constitution enough
to affect its resistance. (We shall later see that the situation with
many of the metals is almost as bad.) Most of the experiments
reveal a steady decline of resistance as the temperature is raised,
whether the sample used be amorphous or crystalline (graphitic)
and whatever its history; but Noyes recently traced several very
concordant curves for several samples of graphite (all however of
the same provenience) showing a minimum of resistance near 800° C.
Diamonds have exceedingly high resistances, which fall when they
are heated.

Passing over four gases and three metals, we come next to silicon ;
the curve traced by Koenigsberger shows the resistance descending
as the temperature is increased, until at a certain critical temperature
it leaps sharply upward; from the new high value it descends again
as the silicon is further warmed, only to make a second upward jump;
from this second maximum it drops steadily away, at least as far as
the highest temperature attained in the experiment. This illustrates
another perplexing property of some elements; they have several
distinct "allotropic" forms, each of them more or less stable over a
distinct range of temperature which may or may not overlap with the
ranges of the others; each must be regarded, so far as its conducting-
power is concerned, as a distinct element. In some instances the
several forms of an element are vividly contrasted in appearance and
in general behavior; such is the case with phosphorus, all of the forms
of which have high resistances, but little is known about their trends
with temperature. In other cases the anomalous changes of temper-
ature with resistance are not accompanied by other striking changes;
and there is a tendency to explain any deviation from an expected
trend — such as, for example, a maximum in a resistance-temperature
curve — by saying that the substance is gradually changing from one
form into another.

Sulphur is the extreme case of high resistance. I know of no data
for scandium, which is to be regretted, as there is some reason from
general atomic theory for supposing that this element stands at a
turning-point of the Periodic Table. Titanium, like silicon, has
several modifications, in some of which the characteristic rises while
in others it descends. Germanium has been studied lately by Bidwell;
it is the element mentioned above which displays a minimum of
resistance at -116° C. Arsenic resembles the metals. Selenium in
the dark has an extremely high resistance; its character when il-
luminated is too much of a subject to be discussed in this place.
Zirconium was found, at least by one observer, to display a minimum

626 BELL SYSTEM TECHNICAL JOURNAL

of resistance at 70° C, though in conductivity it compares favorably
with the accepted metals. Antimony, although ranked among the
metals, is usually to be found among the exceptions to any rules laid
down for them; the same can be said of bismuth. Tellurium is an
outstanding instance of an element with two modifications, and a
sample taken at random is likely to be a mixture of them in un-
predictable proportions, which change when it is heated; the char-
acteristics are correspondingly crooked, and rarely agree. Iodine
has a Very high resistance.

Comparing the metals as a group with the non-metals, the first
striking rule is that their conductivities are much higher and rather
close together; from silver (the most conductive of all substances at
room-temperature), to bismuth, the most resistant of the elements
commonly accepted as metals, the conducting-power descends in the
relatively small ratio of 75 to 1. The next and familiar rule is, that
increasing temperature and increasing resistance always go together;
the characteristic always slants upward to the right, the derivative
and the temperature-coefficient of resistance are positive. It is
customary to say that the resistance is always approximately pro-
portional to the temperature, and that the temperature-coefficient
of resistance always has approximately the one universal value,
which is the value of the temperature-coefficient of volume of an
ideal gas at constant pressure (or its temperature-coefficient of pres-
sure at constant volume). That is to say, when the temperature of a
piece of metal is increased by a given amount, its resistance increases
approximately in the same proportion as would the pressure of a
fixed quantity of an ideal gas, enclosed in a non-expanding container
and raised from the same initial to the same final temperature as the
metal. Were these statements literally true, all the resistance-
temperature curves for metals would be straight lines intersecting the
axis of temperatures at absolute zero. But the second statement
cannot even be considered a good approximation, unless one is willing
to confer the title "good approximation" on a numerical value .00365
which is expected to agree with a set of observed values which ranges
upwards to .0058 (potassium) and .0063 (iron). (I refrain from giv-
ing a lower limit for the range, for a reason which will presently be
made clear.) Also the characteristic curves are not rigorously
straight lines, although it is not unreasonable to call some of them
approximately straight, when one considers how wide is the interval
of temperature over which some of them have been traced. In
some cases a quadratic term added to the linear expression, resulting
in a formula R = R -\-at-\-bt, 2 is sufficient to express the data. Usually,

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 627

but not always, the extra coefficient b is positive; the characteristic
is concave upward. "Usually but not always" is a phrase much
in demand when one is laying down rules for conducting bodies.
In this case metals of the platinum triad furnish the exceptions.
In other instances cubic terms must be added to the formula?, and
in still others even these are inadequate. One of the longest charac-
teristics ever traced, the one determined by Worthing and Forsythe
for tungsten from 1400° to 3250° C, conforms to the equation
R=const. T 1 - 2 .

All these details about values of resistances and shapes of resistance-
temperature curves are sedate and commonplace enough; but there
is one quite extraordinary phenomenon in this field, one of the strange
discontinuities which appear here and there in the theatre of nature
and contribute more of dramatic interest to the spectacle than any
amount of smooth correlations between continuous variables. Ex-
tensions of the characteristics downwards toward the absolute zero
have to follow upon improvements in the art of producing and main-
taining very low temperatures; and for the last twenty years the
advances in this art have been made in the Cryogenic Laboratory
of the University of Leyden, and there the curves have been extended
downwards step by step as additional ranges of cold were made
accessible. The temperatures down to 14° K. attained with liquefied
hydrogen did not affect the resistances of metals in any very startling
way, although the characteristics are generally more sharply curved
there than at ordinary temperatures; but when with the aid of liquefied
helium Kamerlingh Onnes penetrated to within five degrees of the
absolute zero, something astonishing took place.

Kamerlingh Onnes had been experimenting with platinum wire,
and he had found that over the interval of temperatures newly made
available, the interval from 4.3° to 1.5° K. (a small range when meas-
ured in degrees, but a great one when considered in terms of the dis-
tance between its lower limit and the absolute zero) the resistance of
the- wire did not change. This he thought might mean that the proper
resistance of the metal had become exceedingly small, leaving as
the chief component of the observed resistance a term unaffected by-
temperature and due possibly to some such thing as discontinuities in
the wire, for example between the platinum and bits of impurities
mixed into it. To have a purer metal he replaced the platinum by
repeatedly-distilled mercury. It was contained in a slender glass
capillary tube, forming so fine a filament that the resistance at room-
temperature was rather considerable; in one specified instance, 173
ohms. When he lowered this filament of mercury to the temperature

628 BELL SYSTEM TECHNICAL JOURNAL

of frozen helium, at a certain point the resistance suddenly vanished.
Literally it vanished; the word is justified, for the value to which it
had dropped was, if not truly zero, at all events not so much as one
five-billionth of its value at room-temperature, and not so much as
one ten-millionth of its value just before, at about 4.1° K., it suddenly
disappeared. The mercury had altogether lost what had always
seemed to be as inseparable a quality of matter as its inertia or its
weight.

A few other elements were later found to share this property;
tin, of which the resistance vanishes at 3.78°; lead, having its thres-
hold-temperature at 7.2°; thallium, at 2.3°. Three of these four are
consecutive in the procession of elements. Other elements were
definitely found not to become "supra-conductive" within the ac-
cessible range: gold, cadmium, platinum, copper and iron. In the
vicinity of the absolute zero each of these metals has a constant
resistance independent of temperature. This as I mentioned was
interpreted to mean that these metals, or at least these samples,
behaved thus because they were impure — that impurities prevented
the vanishing of resistance — but since mercury contaminated in-
tentionally with gold or with cadmium was found to become supra-
conductive, and tin amalgam likewise, it has become necessary to
save this interpretation, if at all, by assuming that in the five specified
metals the impurities coalesce with the metal in some particular way.
It is interesting to note that the threshold-temperature of tin amalgam
lies above that of either of its components — at 4.29° K., to be com-
pared with the 4.1° of mercury and the 3.78° of tin. These thres-
holds are not entirely independent of circumstances; they diminish
when a large current-density is used, and also when a magnetic field
is applied, possibly from the same reason in both cases.

A number of fantastic things could happen in a world from which
electrical resistance had vanished, and one of them was actually
realized by Kamerlingh Onnes within the compass of his helium-
cooled chamber, when a current of three hundred and twenty amperes
flowed for half-an-hour around and around a leaden ring with no
applied E.M.F. whatever to maintain it, and did not lose as much as
one one-hundredth of its initial strength. In another experiment
a current of forty-nine amperes flowed for an hour around a coil of
lead wire of a thousand turns, wound upon a brass tube, and did not
lose quite one per cent, of the intensity with which it had been started
by removing a magnet of which the field had interlaced the coil.
At this rate it would have taken over four days for the current to
drop to the 1/eth part of its initial value, if the coil could have been

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 629

kept cold so long. This corresponds to a resistance lower than
3.10 -7 ohms; the resistance of the coil at room-temperature was
734 ohms. Few discoveries in physics can have been so exciting as
this one, and further news from Leyden is awaited with keen antici-
pation. Until the present liquefied helium has been made nowhere
else, but from now on the process will be carried on at Toronto also.
Pressure affects the resistance of a metal much less than tempera-
ture; that is to say, doubling the hydrostatic pressure upon a metal
makes no perceptible difference with its resistance if the initial pres-
sure is one atmosphere or less, and usually alters it only by a few per
cent, if the intial pressure amounts to thousands of atmosphere. The
art of applying enormous pressures under controllable conditions has
been developed furthest by Bridgman in the Physical Laboratory
of Harvard University, which through his work holds the same unique
rank in high-pressure investigations as Kamerlingh Onnes' laboratory
at Leyden in low-temperature research. The highest pressure which
Bridgman has applied to metals during resistance-measurements
exceeds 12,000 kg/cm 2 , which amounts practically to twelve thousand
atmospheres. No one has ever applied temperatures twelve thousand
times as great as room-temperature, nor even four thousand times
as great as the lowest accessible temperature; but when the pressure
is altered in this enormous ratio the resistance changes only by a few
per cent. The volume likewise changes by only a small fraction,
which rather suggests that it is the change in closeness of packing
of atoms rather than the creation of intense internal stresses which is
responsible for the change in conductivity; however, there is no close
correlation between relative change in volume and relative change
in resistance; sometimes the two are of opposite signs. Usually, but
not always, the conductivity increases with the pressure; as if squeez-
ing the atoms together facilitated the flow of electricity across the
metal. The rule applies to thirty-five elements, distributed as
follows in the Periodic Table: in the first column, 11 Na, 19 K, 29 Cu,
47 Ag, 79 Au; second column, 12 Mg, 30 Zn, 48 Cd, 80 Hg; third,
13A1, 31 Ga, 49 In, 81 Tl; fourth, 6C, 22 Ti, 40 Zr, 50 Sn, 82 Pb;
fifth, 15 P, 33 As, 73 Ta; sixth. 42 Mo, 52 Te, 74 W, 92 U; seventh, 53 I ;
eighth, 26 Fe, 27 Co, 28 Xi, 45 Rh, 46 Pd, 77 Ir, 78 Pt; rare earths, 57
La, 60 Nd. Several of the non-metallic elements are found in the list.
The exceptions are the five curiously assorted metals 3 lithium, 20
calcium, 38 strontium, 51 antimony, 83 bismuth — five elements
distributed over three columns of the Periodic Table, each of which
contains several other elements which conform to the rule. One
modification of 55 caesium belongs under the rule, another, among

630 BELL SYSTEM TECHNICAL JOURNAL

the exceptions. This illustrates how the behavior of metals in con-
ducting electricity is liable to cut across the classification of the
Periodic System, which controls nearly all of the properties of elements
except those that vary uniformly from one element to the next all
along the series.

As for the magnitude of the effect, the resistances of most metals
are decreased through less than 10% by applying a pressure of ten
thousand atmospheres, some only through one or two per cent.;
but the decrease is 40% for sodium, 70% for potassium, 70%, also
for the "debatable" element tellurium, and 97% for black phos-
phorus; bismuth gains about 25% in resistance and antimony about
10%. The curves representing resistance as function of pressure
are somewhat curved, but not greatly so; however the curvature
frequently varies along the curve to such an extent that a two-constant
formula is not sufficient to express the data. It is an interesting
fact that the percentage by which a given pressure changes the re-
sistance of a metal is approximately independent of its temperature,
and consequently the percentage by which a given rise in temperature
changes the resistance is approximately independent of the pressure;
so that the combined effects of a pressure-change Ap and a temperature-
change AT on a metal change its resistance from R n to R u (1+aAp)

(l+ftar).

Tension, which is equivalent to negative pressure acting along a
particular direction (there is no way of applying a negative hydro-
static pressure) results in lengthening the metal along one direction,
shortening it along all directions perpendicular to that one, and
dilating it as a whole. Most of the information about what it does
to electrical resistance is owed to Bridgman. Usually, but not always,
tension increases the resistance to current-flow along the direction
of the stress. The exceptions are bismuth and strontium. Com-
paring the data about the effects of pressure and of tension, we see that
Bi and Sr are exceptions to the common rules for both, while Li, Ca
and Sb are exceptions to the usual rule for pressure but not to the
usual rule for tension. This helps to show why it is so difficult to
set up a thoroughly satisfactory theory of conduction in metals.

By melting a substance its density can be altered without altering
either its temperature or its pressure; of course, the balance of inter-
atomic forces is also altered in some mysterious but very potent way.
Melting a solid usually brings about a decrease in density; the solid
sinks in the liquid; but there are exceptions (bismuth, antimony,
gallium). The conductivity always changes in the same sense as the
density; hence for most metals the solid is more conductive than the

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 631

liquid, l>ul bismuth, antimony, and gallium have greater resistances
frozen than molten. This is one of the few rules in this field to which
no exceptions have yet been discovered. The observed values of the
ratio (resistance of liquid) (resistance of solid), when tabulated and
examined, show a tendency to cluster about values which are ratios
of simple integers, such as 2:1, 1 :3, 1 :4. It would probably require
a careful and expert analysis to show whether this tendency is more
pronounced than a quite random distribution might reasonably be
expected to display. Mercury has the highest ratio of all, 4:1.

Other agencies which are harder to measure or control may have
distressingly great effects on the conductivity of a metal. The various
metallurgical processes, annealing, cold-working and the rest, affect
the resistance; sometimes the sign of the change can be explained by
saying that the process has caused the many small crystals forming
the metal to fuse into a few large ones, diminishing the resistance
offered by the intercrystalline partitions; sometimes this explanation
fails to work. Impurities may have a serious effect; for example
Bridgman remarks of bismuth that "a fraction of a per cent, of lead
or tin may change the temperature-coefficient from positive to nega-
tive and increase the specific resistance severalfold." Often im-
purities betray themselves by an abnormally low temperature-coeffi-
cient of the metal; this means that the absolute rate of increase is
unusually small compared to the value of the resistance itself. This
is so generally the case that a value of temperature-coefficient which
(at 0° C.) is much below, say, .004 is usually taken to mean that the
sample of metal under investigation is impure; and the "standard"
values for individual metals set down in tables have often taken
sudden jumps upward, when better-purified samples became avail-
able for measurements. For this reason I laid more stress, in a
preceding paragraph, on the values which far exceed .00365 rather
than the values which fall far below it. A metal contaminated by
a small admixture of another metal may be regarded as the limiting
case of an alloy. There is an enormous literature of the electrical
behaviour of alloys, and some of the results can be extended to this
limiting case. It is found, for example, that if two metals A and B
form mixed crystals with one another, an alloy formed by mixing a
small percentage or a fraction of one per cent of A into B, has a sur-
prisingly greater resistance than B; and vice versa. The temperature-
coefficient of the alloy is on the other hand much smaller than that
of the metal, and may even be negative. Thus, although an alloy
of this type may seem to be as thoroughgoing a metal as either of the
pure elements of which it is made, it has a thoroughly anomalous

632 BELL SYSTEM TECHNICAL JOURNAL

electrical behaviour; and the alloys as a whole, instead of assisting us
to understand conduction in metals, contribute generously to the
already abundant supply of difficulties. It remains to be seen whether
the measurements upon single crystals of metals, which arc being
published at a steadily-increasing rate, are going to clarify the sit-
uation or increase the perplexity.

While I have left unmentioned a large number of the phenomena
which a theory of conduction must be required to explain, the few
which I have described will give quite an adequate basis for begin-
ning a discussion of some of the extant theories. It must be conceded
at once that the situation is bad. Perhaps there is some set of assump-
tions or of postulates by which the whole chaotic crowd of phenomena
can be unified into a harmonious system; but if so, no one has yet
formulated it. The theories, such as they are, may be divided into
two groups: theories in which the electrons are supposed to move
freely within the atoms and be stopped when they reach an inter-
space, and theories in which the electrons are assumed to move freely
within the interspaces and be stopped when they collide with atoms.
Those of the first kind start out with the advantage of being better
adapted to the usual effect of pressure on resistance; most metals
become more conductive when compressed, as if conduction were
assisted by squeezing the atoms closer together. Still the oldest,
the best-known, and the most highly elaborated of all the theories
belongs to the second kind. This is the one formally known as the
electron theory of metallic conduction, or more briefly as the electron
theory of metals, and quite commonly as the "classical" theory of
conduction (it does not take an idea so long to become "classical"
in physics as it does in the arts). Founded by Riecke and by Drude
in the closing years of the last century, it was developed by Lorentz and
has since been worked over by Planck, Wien, Bohr, and other savants
of the first eminence. Its popularity is largely due, I suspect, to the
fact that it can be formulated with great if specious exactness: that is
to say, as soon as a few definite assumptions are made (such as the
simple, if implausible, assumptions that the atoms are big elastic
spheres and the electrons little ones), numercial consequences can be
calculated with any degree of precision. In this respect most of the
competing theories are sadly defective. Two or three of the numerical
deductions made from simple auxiliary assumptions have agreed
rather well with experimental data; and they have contributed to
the feeling that there must be some kernel of truth in the mathematics,
even if not in the physics of the thing, although it breaks down in so
many other comparisons with experiment.

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 633

Fundamentally the theory is very simple, and has not been helped
to any great extent by the more sophisticated mathematics which its
emendators have introduced into it. What is observed in electrical
conduction is this: when a potential-difference is established across
a piece of metal, the electrons do not fall freely clear across it and
emerge at the positive end with all the kinetic energy which the P.D.
should have communicated to them; they ooze gradually through the
metal, heating it as they go along and emerging with no unusual
amount of energy, as if they had rubbed along through the metal
like heavy particles dropping at constant speed through a gas. "Rub-
bing along" being a concept foreign to the atomic scale, we have to
interpret that each electron falls freely through a small distance,
collides with something to which it gives up the energy acquired from
the field during its fall, falls again across another short distance,
gives up its new quota in another collision, and so forth from side to
side of the metal. Furthermore the energy which it gives up at each
stoppage must find its way directly or indirectly into the heat of the
metal, i.e., into thermal agitation of its atoms. Representing by T
the time-interval between two consecutive collisions, by E the field-
strength in the metal, by e and m the charge and mass of the electron,
by U the average kinetic energy acquired by the electron from the
held in its free fall between two collisions, we have

U=l(eET/m) -m. (1)

If there are n electrons in unit cube of the metal, and each is stopped
1/7' times in unit time, the rate at which heat appears in the unit
cube is nU/T; but this rate is by definition the product of the con-
ductivity <r by the square of the fieldstrength E, hence

<r = h>ie 2 T/m (2)

The same equation (2) can be reached, if one prefers to think of
conductivity as the ratio of current-density to fieldstrength, by con-
sidering that during each free fall, the field augments the speed of
each electron in the direction of the field-vector by the amount eET/m,
which on the average is lost at the collision terminating the fall; so
that the result is as if the field imprinted a constant drift-speed equal
to \cEf m upon all the electrons. Multiplying by ne to get the current-
densitv and dividing by E to get the conductivity, we arrive again
at (2).'

Equation (2) is the fundamental equation of the electron theory of
conduction, and indeed of most of the other theories. Let us begin
by trying the supposition that the electrons are at rest until the field

634 BELL SYSTEM TECHNICAL JOURNAL

is applied, and arc brought to a full stop at each collision. Represent
by / the average distance traversed between collisions. The pro-
posed assumption leads to T =v'2ml/cE. The conductivity therefore
would depend on the fieldstrength, which would violate Ohm's law.
Ohm's law being rigorously valid except under extreme conditions
(Bridgman found the first slight deviations from it, in gold and silver,
at current-densities of the order of 10 fi amps /cm-) we have to discard
the idea. The lesson is, that the electrons must be supposed to be
normally in motion at speeds enormously greater than the speed
imparted by the field during a free fall. Let u stand for the natural
average speed of the electrons; we have T = l/u, and

a = \neH/mu, (3)

provided always that ifyeET/m.

This condition is abundantly fulfilled if we make the obvious and
appealing assumption that the electrons are moving with the same
average kinetic energy as atoms of a gas at the same temperature;
in fact, if the free path / is no longer than the average distance be-
tween atom-centres, the deviations from Ohm's law should not appear
even under such extreme circumstances as those of Bridgman's ex- ■
periments. Making therefore this assumption, which in symbols is

3

hnu 2 = -^-kT, we find

a = \-4= * (4)

Not much attention should be paid to the numerical factor, which would
be slightly different if we should assume Maxwell's law of distribution
for the velocities of the electrons; the essential factor is the last one,
nl/y/T . Examining (4) in the light of the fact that the conductivity
of most metals decreases distinctly more rapidly than 1/v 5 n = in fact,
as rapidly as 1/Tor still more so — as the temperature increases, we
see that the product nl will have to be supposed to vary with temper-
ature. It seems natural to suppose that I depends altogether on the
distance between adjacent atoms, which varies comparatively little
with temperature, and anyway varies in the wrong direction for the
purpose of the theory; so that the burden of accounting for the pro-
portionality of <t to the first or a higher power of 1/7" must be laid
upon n.

Now it has occurred to a number of people that the free electrons
are dissociated from the atoms, and the number of free electrons

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 635

is given by i lie derive nf dissociation, which in turn should vary
with the tempera I urc in a manner prescribed altogether by the amount
of work necessary to remove an electron from an atom into I lie
(presumed) interspace where it plays about freely, lint we should
certainly expect that this work would be positive, as it is for the
extraction of electrons from free atoms; in which case the degree of
dissociation and the number of free electrons should increase with
temperature. The theory is therefore adapted to explain a resistance
which decreases steadily with increasing temperature, as do the
resistances of some non-metallic elements; it is adapted to explain
a resistance which at first diminishes and then, as the temperature
increases further, goes through a minimum and rises, for the decrease
in the factor / ' V T finally predominates over the increase in the factor
11 ; it is not adapted to explain a resistance increasing with temperature
over the whole range, as do those of the metals. One might assume
that the work of extracting an electron from an atom inside the
metal is negative. This is essentially the alternative embraced by
Waterman, who postulates that the work in question is a function of
temperature, of the form W=W — cT, f>0. For metals W„ is to
be chosen negative or zero, so that W shall be negative throughout;
for non-metallic elements W is to be given some positive value, so
that W shall change in sign at some point in the temperature-range.
This unusual theory must be judged by its effectiveness; that it
should reduce conduction in all elements, metallic and non-metallic
alike, to a phenomenon of a single type is a feature appealing strongly
in its favor; but Noses' curves of resistance versus temperature for
graphite did not agree with its demands in a satisfactory manner.

The assumption underlying (4) has however involved us in a col-
lateral difficulty. If we believe that the ;/ free electrons per cc. of

the metal have an average energy \-rkT and a total kinetic energy
—nkT, we are certainly forced to admit that when the unit cube of

metal is heated through 1° the electrons must take their share -^r- nk

of the heat imparted to it; but the specific heat of most metals is
such that it seems that the atoms must take it all and leave none
over for the electrons. If we evade this difficulty by assuming //
to be quite small compared with the number of atoms per cc, a few
per cent, of it or less, we lose certain numerical agreements which
will be mentioned later, and we have also to make / quite large,
amounting to several times the distance between adjacent atoms;

636 BELL SYSTEM TECHNICAL JOURNAL

yet all the tendency of modern atomic theory is to make it seem likely
that the atoms fill almost the whole space within the metal.

Another way to avoid the difficulty with the specific heats consists
in assuming that the high natural speed with which the electrons
fly about is altogether independent of temperature; the burden of
making a as expressed in (3) vary in the proper manner with temper-
ature is then laid upon /, which, Wien suggested, should be supposed
to vary inversely as the amplitude of vibration of the atoms — that
is, a free electron collides with an atom only if and when it is in vibra-
tion, and the chance of a collision increases with the amplitude of the
vibration. The variation of resistance with pressure may then be
explained, so far as the usual sign goes, by saying that when an ordi-
nary metal is compressed the amplitude of oscillation of its atoms
diminishes, though the temperature remain the same; the frequency
of oscillation must then vary inversely as the amplitude, to keep
the average energy of oscillation constant; there is some reason for
expecting this to happen. Bridgman's theory somewhat resembles
this one, except that the electrons are supposed to glide through the
atoms and collide with the gaps; gaps between atoms are compara-
tively unusual, and occur chiefly when two atoms are vibrating with
great amplitudes in opposite senses, so that the variation of con-
ductivity with pressure again has the proper sign. But to explain
the behavior of the three metals of which the resistance increases with
pressure and with tension, Bridgman went back to the idea that in
these the electrons glide through the interspaces.

As I have given only the phenomena of conduction which the
electron-theory explains with difficulty, I must in justice mention
the ones on which its reputation chiefly depends. In the first place
it is a theory of thermal conduction as well as electrical conduction;
the electrons in the hotter part of a metal maintained at an uneven
temperature are assumed to have a greater average energy than the
electrons in the cooler part, so that they diffuse down the temperature-
gradient and realize a convection-current of heat. The theory leads
to as definite a numerical value of the one conductivity as of the
other, and the ratio of electrical to thermal conductivity is predicted as

a universal constant for all metals, multiplied into the absolute tem-
perature, and devoid of the quantities n and I which have caused us
so much trouble. This is one of the predictions which is nearly
enough true to be impressive; the ratio X/oT does indeed vary sur-

SOME CONTEMPORARY ADVANCES IN PHYSICS— J' 637

prisingly little over a wide range of metals at room-temperature and
over a fairly wide range of temperatures for each of many metals.
It is usually somewhat larger than the predicted value (5); but this
can he conveniently explained by saying that there must be an addi-
tional mechanism for transmitting heat, something in the nature of
the elasticity of the substance, which superposes its conducting-
power upon the conducting-power of the electrons, and so inflates
the numerator of the ratio in (5). The reason for supposing such an
extra mechanism is primarily that there must be some such mechan-
ism to perform the thermal conduction in substances which are
electrical insulators. No element conducts heat as badly as sulphur
and boron conduct electricity; and if we imagine a special elastic
mechanism for conducting heat in boron and sulphur, we can hardly
deny it to copper and silver. Bridgman found that for six metals
out of eleven tested, the thermal conductivity decreased when high
pressure was applied, although the electrical conductivity increased.
We must hope to find an explanation for this anomaly in the behaviour
of the elastic mechanism; likewise an explanation for the deviations
from (5) which occur at high and at low temperatures. In theories
such as the one mentioned over Wien's name in the last paragraph,
in which the average vis viva of the electrons is supposed not to vary
from a hotter place in a metal to a cooler place, we have to lay the
entire burden of thermal conduction upon the elastic mechanism.
This makes it difficult to explain the universal relation (5).

Another striking feature of the theory is that Lorentz succeeded
in deducing the Rayleigh-Jeans radiation-law from it. He obtained
from it an expression for E, the radiant emissivity of a thin stratum
of metal, as a function of temperature T of the metal and wavelength
X of the radiation; another for A, the absorbing-power of the metal,
likewise a function of T and X; divided the first by the second,
and obtained a definite quotient. By Kirchhoff's thermodynamic
laws, E/A is equal to E , the radiant emissivity of a perfectly black
body. The expression deduced by Rayleigh and by Jeans for E
and the expression deduced by Lorentz for E/A are identical. Lorentz
assumed that the collisions of the electrons with the atoms (or what-
ever it is they collide with) are very short in duration compared with
the intervals of free unaccelerated flight from one collision to the
next, and that the speeds of the electrons are distributed according
to Maxwell's law about the mean value corresponding to the mean
energy SkT/2. He also made certain assumptions which restrict
the validity of his expression for E/A to radiations of great wave-
length ; the Rayleigh-Jeans expression for E is restricted in exactly

638 BELL SYSTEM TECHNICAL JOURNAL

Lhe same way. At least as much, it seems, should be demanded
from any theory of conduction offered in competition with the "clas-
sical" one.

The conception of free electrons in metals also gives a beautiful
qualitative explanation of the thermoelectric effects, although un-
fortunately it does not do very well as a quantitative theory. If
in two metals at a certain temperature the densities of free electrons
are different — ri\ free electrons per cc. in one and n* in the other —
and these two metals are brought into contact with one another,
electrons will flow from the one where the density is greater into the
one where it is less; and this flow will continue until arrested by a
counter-electromotive-force V, of which the equilibrium-value can
be shown, in any one of a variety of ways, to be

kT
V= — Inhh/ni)
e

Such an electromotive force would account for the Peltier effect;
and conversely, if the theory were correct, measurements of the
Peltier effect between two metals at a given temperature and pres-
sure would give the ratio between the densities of free electrons in
the two metals under the specified conditions. Such data, combined
with data on conductivity interpreted by such an equation as (4),
should give information about the free paths h and k in the metals.
The Thomson effect is more difficult to deal with, as thermal equi-
librium does not prevail; however it can be .seen that there will be
a counter E.M.F. in an unevenly-heated metal. Measurements on
the Peltier and Thomson coefficients for many metals, over wide
ranges of temperature and pressure, would be very valuable; but
they are so extremely hard to make even under the best of conditions,
that the outlook for obtaining a really extensive set is unpromising.
Possibly there is a better chance with the indirect method (detei min-
ing the first and second derivatives of the curve of thermal electro-
motive force versus temperature). Such data of the Thomson effect
as exist are not helpful to the simple theory.

Another phenomenon which lends itself very readily to explana-
tion by the theory, and so contributes a certain amount of support
to it, is the thermionic effect — the spontaneous outflow of electrons
through the surfaces of hot metals. (But carbon likewise exhibits
it very efficiently, and we must beware of formulating any theory of
it which reposes on specific properties of metals not shared by carbon!)
To interpret the thermionic effect only one new feature need be added
to the theory, and this a feature which in fact was all the time latent

SOME CONTEMPORARY ADVANCES IN 1'IIVSICS— V 639

in it — the idea that there is a certain fixed potential-difference between
the interior of a metal and the region outside of it, resulting in a
potential-drop localized in a thin stratum at the surface, which an
electron within the metal must surmount in order to escape from
the metal into a contiguous vacuum. Such a potential-drop would
for instance result from a "double layer" along the surface of the
metal, a sheet of positive charges within and a sheet of negative
charges opposite, parallel, and close to the positive sheet on the
outside. It has been pointed out that, since probably half of the
orbital electrons belonging to the atoms at the frontier of a metal
lie outside the plane containing the nuclei of these atoms, they with
the nuclei constitute a sort of double-layer; it has also been suggested
that after a certain number of electrons issue from the metal, they
are held as an electron-atmosphere above it by the forces due to the
distribution of residual positive charge within the metal (Kelvin's
electrical-image conception), and the electron-atmosphere with t he-
positive surface-charge together form a double-layer. However we
may conceive this double-layer, it is obvious that if we postulate
free electrons within the metal, we must also postulate a barrier
in the shape of an opposing potential-drop between the metal and
the exterior world to keep the electrons from wandering away.

Designate this potential-drop by b, so that eb is the energy which
an electron must give up in traversing it from inside to outside.
Assume further (disregarding the old specific-heat difficulty) that
the velocities of the electrons inside the metal are distributed iso-
tropically in direction, and according to Maxwell's distribution-law

in speed, with the mean kinetic energy— kT appropriate to the tem-
perature T of the metal. Imagine the metal surface to occupy the
plane .v = 0, metal to the left and vacuum to the right. Consider
the electrons which come from within the metal and strike unit area
of the boundary in unit time; those of them which have velocities
of which the ^-component lies between it and u+du are in number
equal to

(11= —=e 2kT(ln, (6)

V 2irkT/m

n meaning as heretofore the number of electrons per unit volume of
metal. The total number which strike unit area of the boundary
from within is equal to the integral of this expression from it=0 to
ii = '■*> , which is

I = n\/Frr2irm. (7)

640 BELL SYSTEM TECHNICAL JOURNAL

Those which escape are those for which \mu 2 excee ds eb; w e obtain
the number of them by integrating (6) from u = \/2eb/m to M = oo ,
and find

I, = n VkT/2irm e ~ •■ b ' kl '. ( 8 )

This, supposing n and b to be independent of temperature, is Rich-
ardson's well-known formula for the saturation-current from a hot
body as function of temperature. All of the multitudinous obser-
vations agree with it; but this does not mean so much as might be
thought, for the experts inform us that all the data, no matter how
accurately taken, would agree quite as well with a formula in which
T, or T 2 , or even 7°, stood in the place of the factor T H by which
the exponential is multiplied. Incidentally this would permit us to
make n vary as some small power of temperature, such as the inverse
square root, if we chose to make the resistance-temperature relation
in (4) agree with experiment at such a price. Or if we assume n
independent of temperature, we can calculate it from measurements
on thermionic saturation-currents. The measurements usually give
for n values of the order of magnitude of the number of atoms per unit
volume.

What is more definitely significant is, that the velocities of the
emerging electrons are actually distributed in a manner compatible
with the assumptions made. Let us enquire how many of the electrons
issuing from unit area of the metal have velocities of which the x-
component lies between u and u+du. These are the very same
electrons which struck the surface from within, having velocities of
which the ^-component lay between u' and u'+du'; u' and du' being
related to u and u+du by the equations:

\mii 1 +eb = \m{u')\ u'du' = udu. (9)

The number of these electrons is by (6)

nu> J^Sf Jmtl (10 )

y/2irkT/m

which by virtue of the relations (9) reduces to

e kf- e 2kTdu, \ LL >

>/2irkT/m

which is identical with (6) except for a constant factor; which means
in turn that the distribution-function of the emerging electrons is
identical with the distribution-function of the internal electrons,

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 641

being in fact the Maxwell distribution-function with the same mean

3
kinetic energy— kT. The argument as given proves the point only

for the distribution in the velocity-component u; but the distribu-
tion-functions in v and iv, the velocity-components parallel to the
boundary of the metal, are unaffected by the double-layer, since v
and w for any particular electron are unaffected by the passage
through it; and since it is the essential feature of the Maxwell dis-
tribution-law that the distributions in v and w are identical for each
and every value of u, the conclusion follows as stated. Nevertheless
it does sound paradoxical.

This conclusion has been verified repeatedly by experiment.
Richardson began by simulating the simple mathematical conditions
of infinite plane electrodes as closely as practicable; he inserted a
small flat incandescent surface in an aperture in the middle of a large
flat cold plate, charged the two to the same potential, and placed op-
posite and parallel to them a large flat collecting-electrode. Charging
this latter to various potentials V inferior to the potential of the emit-
ting surface, he plotted the electron-current which it received as func-
tion of V; this is the distribution-function of the speed u of equation
(6) and the following equations translated into terms of the correspond-
ing kinetic energy \ mu 2 as independent variable. To ascertain the
distribution-functions in v and w he isolated a small area of the collec-
ing-electrode, moved it to and fro in a plane parallel to the plane of the
emitting surface, and measured the current into it in its various posi-
tions. Many measurements have since been made upon the currents
into cylindrical collectors from hot wires stretched along the axes of
the cylinders; it is somewhat more difficult to write out the formula
for the expected relation between current and retarding-potential, but
the experimental conditions are much more under the experimenter's
control. All these investigations have confirmed the theorem, except
a single discordant one which was later explained away; the strongest
verification is furnished by the experiments of Germer, whose pre-
cautions of preparation and accuracy of measurements far surpassed
everything that had gone before.

The evidence thus is quite favorable to the idea of an electron-
gas within the metal with its electrons moving with velocities as
prescribed by Maxwell's distribution-law, and kept from diffusing
away by a double-layer covering the surface. Other evidence for
the existence of a double-layer is furnished by the photoelectric
effect and by the existence of contact-potential-differences. When

642 BELL SYSTEM TECHNICAL JOURNAL

light of frequency v falls upon a metal, electrons emerge from it with
velocities which are distributed in a manner quite distinct from
Maxwell's distribution and have nothing to do with the temperature
of the metal. The kinetic energies of some of the electrons attain a
certain upper limit W m , but none surpasses it; W,„ is a linear function
of v given by the equation

W m = hv-P, (12)

//. being Planck's constant, P a positive constant characteristic of the
metal. This is an exceedingly strong intimation that each of the
emerging electrons, while still inside the metal, suddenly absorbed
a quantum of energy hv from the light and departed with it, giving
up a fixed quantity P in passing through the surface. (Those which
issue with energies clearly less than W„, can be supposed to have
started distinctly beneath the surface and to have lost additional
energy in struggling through the metal to it). Translating P into
potential-drop, we see that it represents the potential-difference or
the "strength" of the surface double layer. It may be determined
by measuring W m for light of various frequencies, plotting it against
frequency, and extrapolating the resulting straight line to its inter-
section with the axis of frequencies. Or it may, in principle, be
determined by plotting the photoelectric current as a function of
frequency, and extrapolating the curve to its intersection with the
axis of frequencies, where no electrons escape and the photosensitive-
ness ceases; but curves are not so easy to extrapolate as straight lines,
and there are some anomalous results which are still unexplained.

It would seem an easy matter to measure the strength of the double-
layer by both photoelectric and thermionic methods upon a single
substance. But it is rather difficult; for one reason, the substances
for which the photoelectric currents are easy to produce and measure
are precisely the metals upon which good thermionic measurements
are next to impossible, and vice versa. The best photoelectric meas-
urements have been made upon the alkali metals, which are very
sensitive to visible light; but they cannot be formed into wires, and
volatilize furiously when heated enough to produce an important
thermionic effect, filling the evacuated tube with dense vapors which
ruin the accuracy of the measurements. The best thermionic measure-
ments have been made upon platinum and tungsten, which are not
sensitive at all to visible light, and begin to be sensitive far out in
the ultraviolet where experiments with radiation are difficult. Fur-
thermore there is the capital difficulty that the photoelectric measure-
ments must be confined to temperatures where the thermionic current

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 643

is imperceptible; if one were to irradiate an incandescent tungsten
filament the extra current of photoelectrons would be too small to
notice. If we assume outright that P does not vary greatly from
room-temperature up to the temperatures of incandescence, and
therefore compare photoelectric data upon cool metals with thermionic
data upon the same metals when hot, we find that there is a fairly
good agreement. Values of the thermionic constant b between 4 and 5
volts correspond to photoelectric sensitiveness commencing between
3,100 and 2,500 Angstrom units, and this correctly describes the
behavior of several of the heavy high-melting-point metals: photo-
electric sensitiveness extending well up into the visible spectrum,
such as the alkali metals display, corresponds to values of P/e of the
order of 2 volts and lower, and such values are indicated by the
thermionic experiments made upon sodium and potassium by Richard-
son under the inevitably bad conditions.

Contact-potential-difference, one of the longest known of all elec-
trical phenomena — Volta discovered it — agrees admirably with this
interpretation of the photoelectric constant P. Imagine that we have
pieces of two metals, potassium and silver for example, which are
drawn out and welded together at one end, and at their other ends
are spread out into plates and face one another across a vacuous
.space. We know that the opposing faces behave as if they were at
essentially different potentials, the potential-difference V between
them being characteristic of the two metals and independent of the
size or separation of the opposing faces. Yet this potential-difference
V is not equal to, is indeed usually much greater than the potential-
difference between the interiors of the metal across the welded joint,
which is deduced from the Peltier effect. The only way to resolve
the contradiction is to assume that it is the region just outside the
potassium which differs by V from the region just outside the silver;
the metals themselves are at nearly the same potential, but there is
a double-layer at the surface of each which establishes a fixed potential-
drop between it and the vacuum. Representing by Pi/e and by Pi' c
the voltage-drops at these two double-layers, by .1/ the potential-
difference between the interiors of the metals as inferred from the
Peltier effect, by V the potential-difference between the regions just
outside the metals which we identify with the contact potential
difference, we find

p.,'e-P l 'e=V+M (13)

in which M is so small compared with the other terms that hence-
forth we will leave it out

644 BELL SYSTEM TECHNICAL JOURNAL

Now imagine that light of a high frequency v falls upon the potas-
sium; it elicits electrons of which the maximum energy at emergence
is hva — Pi] these highest-speed electrons arrive at the silver plate
with energy (hv — P\/e—V), having had to overcome the additional
potential-drop V in passing from the region just outside the potassium
to the region just outside the silver. (The reader can make the changes
in language required if V happens to be of the sign corresponding to
a potential-rise). From (13) we see that this energy of arrival is
equal to (hv — Pi/ e) — an expression from which Pi, the only quantity
characterizing the irradiated metal, has fallen out! Therefore the
electrons arrive at the silver plate with the same maximum speed,
whether the irradiated metal be potassium, sodium, silver, or any
other metal! (unless we hit upon a metal for which hv <Pi/e, in which
case we shall never get any at all).

This experiment is usually performed by putting a battery between
the silver and the irradiated metal, and adjusting its E.M.F. until
the fastest electrons are just turned back before reaching the silver;
this is known as "determining the stopping potential." If our inter-
pretation of contact-potential-difference is correct, the stopping-
potential must be independent of the irradiated metal, and depend
only on the material of which the collecting-electrode is made; further,
the difference between the stopping-potentials observed with two
different metals as collecting-electrodes should be equal to their
contact potential difference. These predictions have been verified
in several sets of experiments, notably by Richardson and Compton.
Millikan developed the interesting theoretical consequences which
they suggest. There should be similar relations involving thermionic
currents; observations confirming them have been made, but not so
extensively published; they are more difficult to make with accuracy
because the thermionic electrons have no definite recognizable maxi-
mum velocity.

We seem to have marshalled a formidable amount of evidence in
favor of the electron-theory of conduction with the associated idea
of the surface double-layer. Yet it would be misleading not to point
out that an equation quite as satisfactory as (8) in representing the
thermionic current as function of temperature can be deduced by
reasoning in an entirely different fashion from entirely different
postulates. This, the thermodynamical method of speculating about
the thermionic effect, was originated by H. A. Wilson; it consists
essentially in assuming a thoroughgoing analogy between the outflow
of electrons from a hot metal and the evaporation of molecules from
a solid or a liquid. We know that if an evacuated chamber is partly

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 645

filled with liquid water or solid CaC03, the remaining space inside
the chamber is quickly pervaded with H 2 or CO2 molecules com-
posing a gas, its pressure and density being determined absolutely
by the temperature T. We infer that if an evacuated chamber,
with its walls made of some insulating substance, contains a piece of
metal and is heated to a high temperature, the whole evacuated
space will be pervaded with electrons composing a gas, its pressure
p and density n being determined absolutely by the temperature
of the system, T. We must assume that the electron-gas outside
the metal conforms to the ideal-gas law

p = nkT (14)

and we shall also presently assume that its specific heats have the
values characteristic of monatomic ideal gases,

C v = \Nk, C P = ^Nk. (15)

I use n to represent the number of electrons per unit volume of the
gas, as the number within the metal no longer enters in any way
into the reasoning; N to represent the number in a gramme-mole-
cule (Avogadro's constant). These are the only assumptions which
involve a kinetic theory in any way.

Imagine now a wire of which one end projects into an evacuated
chamber of the sort described, maintained at T, and the other into
another such chamber maintained at T-\-dT. We consider a process
which consists of increasing the volume of the first chamber by just
enough to require N additional electrons to come out of the wire
to fill the additional space, and simultaneously decreasing the volume
of the second chamber by just enough to crowd N electrons into
the wire; so that in effect N electrons are transferred from the one
chamber to the other through a wire of which the two ends are at
temperatures T-\-dT and T. This process will be carried on re-
versibly. Designate by L the heat which must be imparted to the
metal at T, to remove one electron from it under the circumstances
of the experiment; by sdT the heat which is absorbed when one
electron is transferred through the metal from a point where the
temperature is T to a point where the temperature is T-\-dT. s is
the coefficient of the Thomson effect, referred to a single electron
instead of a coulomb. L contains a term kT, which corresponds
to the mechanical work done in forcing back the walls of the chamber
to make place for the evaporated electron-gas. Subtracting it we

646 BELL SYSTEM TECHNICAL JOURNAL

obtain (L—kT), to be called e<f>, as the actual energy expended in
putting the electron across the boundary of the metal.*

In the process which I have just described, the input of heat con-
sists of the following terms: NL which goes to extract the N electrons

from the metal in the first chamber, - N[L-\-^-,dT ) which is liberated

when N electrons condense into the metal in the second chamber,
and -NsdT which is absorbed by the electrons in travelling through
the wire. The output of work is NkT during the expansion of the first
chamber, -NkT-NkdT during the contraction of the second cham-
ber. The input of entropy is NL/T during the evaporation in the

first chamber, ~n(l/T+ d<<L J^ )dT during the condensation in the

second chamber, and (-Ns/T)dT during the flow of electrons through

the wire.

We now complete the cycle by changing the pressure and tem-
perature of the gramme-molecule of electron-gas in the first chamber
from p, T to p-\-dp, T+dT, after which it becomes equivalent with
the gramme-molecule in the second chamber at the beginning of
the process. Calculated in the usual way — isothermal contraction
at T from p to p+dp, isobaric expansion at p-\-dp from T to T+dT

—we find: input of heat, ~-NkdT- NkT[d(ln p)/dT]dT; output of
work, NkdT-NkT[d(lnp)/dT]dT; input of entropy, -^(Nk/T)dT-

Nk[d(ln p)/dT]dT.

The two processes together constitute a complete reversible cycle.
We therefore equate the sum of the inputs of entropy to zero, and
obtain :

/ a J ■£ a J

(10)

and equate the difference of the inputs of heat and the outputs of
work to zero, which gives:

-$-,+.§*-0 (17)

* This definition suggests a thermal method of measuring L, which has several
times been put into practice. The experiments are difficult and the data must
be corrected for many influences, but the best results indicate that (L—kT) is ap-
proximately equal to eb of (8). The data of Darisson and Germer indicate a slight
difference, "which may be an important test of suggested theories oi conduction.

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 64/
and comhiiu' the ccjiiatiinis into

f+k-kT^ (I8 )

which integrated, yields

p=ATe J fcP . (19)

/*«■

We still have to make the bridge between this formula, which
relates to the pressure of the electron-gas in equilibrium with the
metal, and the quantity actually observed, which is the saturation-
current out of the metal surface in an accelerating field. In the
equilibrium-state, the number of electrons which issue from the
metal is equal to the number which, coming from the external electron-
gas, strike its boundary and do not rebound. This is indisputable;
to make it useful we have to make two new assumptions: one, that
the number of electrons which issue from the metal is the same in an
accelerating field as in the equilibrium-state; the other, that no
electrons rebound from the surface. The first assumption had to be
made in the preceding deduction — that is, we had to assums tacitly
that the uncompensated outflow of electrons through the surface of
the metal did not appreciably distort the Maxwell distribution within;
the second is a drawback peculiar to the thermodynamic method.
Accepting these two assumptions along with all their predecessors, we
finally reach the expression for the number of electrons emitted per
unit area per unit lime from the surface of the hot metal :

I=CTXe / kT * . (20)

This is the equation for the thermionic saturation-current attained
by the thermodynamical reasoning.

Let us finally try some hypotheses about the variation of 4> with
temperature: for a first one, the hypothesis </> = constant. The
general equation becomes

I = Cr-ie-W, (2D

which is perfectly identical with (8) which was deduced from the
electron-theory with the additional assumption of a double-layer
independent of temperature. We cannot however freely make an
assumption like this, for our equation (17) shows that an assumption
about def) d'V implies, and conversely is implied by, an assumption

648 BELL SYSTEM TECHNICAL JOURNAL

about the value of the Thomson coefficient s. In making </> inde-
pendent of temperature we in effect assumed that the Thomson

coefficient has the value s = — k (per electron), which happens to be

precisely the value demanded (and vainly demanded) by the electron-
theory of conduction. If on the other hand we choose to accept
from the experiments the fact that 5 is extremely small compared to
q
~k, the equation (16) compels us to set

a

<f> = ^kT/e+<t> . (22)

Inserting this into (19) we obtain

1= CTH'kf, (23)

which is commonly known as the IP-law, and is at the moment the
favorite way of expressing the variation of thermionic current with
temperature. As I said earlier, experiment is thus far powerless to
distinguish between (8), (20) and (22).

This brief and superficial sketch of the thermodynamic argument
is meant partly to familiarize the reader with the P formula, and
partly to show that the observations upon the dependence of ther-
mionic current- on temperature do not necessarily sustain the parti-
cular type of theory which has figured most in these pages, as against
its rivals actual or conceivable. Of course it would be unjustifiable
to say that any argument of the thermodynamical type is ipso facto
stronger than any argument based on a physical model. It may be
true that the laws of thermodynamics are valid everywhere without
exception; but it is certainly true that in any particular case it is
extremely difficult to feel sure just how they should be applied to
arrive at absolutely binding conclusions. In this case, for instance,
we have assumed as both possible and reversible a process which
no one has ever carried through, and no one, in all likelihood, ever
will; and in the course of analyzing the transfers of energy between
the system and the external world in this imagined process, we have
classified some as transfers of heat and some as transfers of mechan-
ical work, and possibly ignored yet others, so that the analysis re-
quires careful thought and has in fact been made in different ways
by different authorities. There is for example the problem of the
allowance to be made for work done in transferring the electrons
from place to place against electromotive forces, which might or
might not be nil when summed around the complete cycle; H. A.

SOME CONTEMPORARY ADVANCES IN PHYSICS— V 649

Wilson has recently made a specific assumption regarding these.
For still further subtleties Bridgman's theoretical articles may be
consulted. I must however add that an extension of the thermo-
dynamical argument, with the assistance of Nernst's "third law of
thermodynamics," leads to the conclusion that the constant C of
equation (23) should have for all elements, if not indeed for all sub-
stances, the same universal value, calculable in terms of certain
universal constants. There is some evidence that this may be true
for emission from pure elements. Were it so, the result would be
of fundamental importance; but another article almost as long as
this one would be required to explain it properly.

The general tone and character of this article will probably leave
the final impression that the electrical behaviour of solids is an utterly
confused and chaotic department of physics, a hopeless entangle-
ment of incongruous rules diversified by numberless exceptions.
I fear that this impression — except perhaps for the hopelessness of
the situation — is substantially the correct one. In fact this pres-
entation has put the state of affairs in rather too favorable a light,
for I have passed over a number of the perplexities. I have scarcely
mentioned the thermoelectric effects, or spoken of the complexities
of the photoelectric effect, or of the emission of electrons from metals
which are bombarded by other electrons or by ionized atoms; and I
have not mentioned at all the galvanomagnetic and thermomagnetic
effects, the most baffling and bewildering of all. In fact it seems
only too probable that if one should succeed in erecting a theory by
which all the phenomena I have described could be brought into
one coherent system, some galvanomagnetic effect would be lying
in wait for it to bring it to the dust. Clairaut is said to have been
saddened by feeling that Newton had discovered all the laws of
celestial mechanics, leaving nothing for men born after him to do
except to improve the methods of calculation. Ambitious students
of physics who, through too exclusive a study of the radiations from
atoms, may have come to feel in the same way about Bohr, should
find consolation in contemplating the present status of the Theory
of Conduction in Solids.

LITERATURE

The chief recent compilation of data upon conduction in solids is Koenigsberger's
article in Graetz' Handbuch der Elektrizitat. K. Baedeker wrote an excellent short
account of the data and the theories, entitled Die elektrischen Erschehiungen in
metallischen Leitern, which although published in 1911 is not yet superseded. Bid-
well's paper on germanium is in Phys. Rev. (2) 19, pp. 447-455 (1922); Noyes'
article on carbon in Phys. Rev. (2) 84, PP- 190-199 (1924). The investigations on
supraconductivity are reported chiefly in the Leyden Communications; Crommelin

650 BELL SYSTEM TECHNICAL JOURNAL

has given a comprehensive account of them in Phys. ZS. 21 (1920), with a bibliog-
raphy of all of the work; two or three subsequent communications arc reviewed
in Science Abstracts. Bridgman's work on the effect of pressure and of tension on
the electrical and thermal conductivities of the elements is printed chiefly in the
Proceedings of the American Academy of Arts and Sciences from 1917 onward, with
occasional announcements in the Physical Review, where also his theoretical papers
are published (Phys. Rev. 14, pp. 306-347 (1919); 17, pp. 161-195 (1921) and 19,
pp. 114-134 (1922). For the effect of melting, consult Bridgman's papers, and
one by von Hauer, in Ann. d. Phys. 51, pp. 189-219 (1916).

The "classical" theory of conduction is presented in Lorentz' book The Theory of
Electrons, which bears his signature as of 1915. Bohr wrote a dissertation upon it
which is highly praised by those who have succeeded in reading it in the Danish.
YVien's and Planck's modifications of it are published in the Sitzungsberichte of the
Berlin Academy for 1912 and 1913. In the Philosophical Magazine of 1915 there
are a number of articles on the theory by G. H. Livens, like Baedeker a victim of
the war. The conception of quantity of free electrons determined by dissociation
of atoms is presented by Koenigsberger in Ann. d. Phys. 82, pp. 170-230 (1910) and
Waterman's extension of it is in Phys. Rev. 22, pp. 259-270 (1923). Some chapters
in J. J. Thomson's Corpuscular Theory of Matter deal with the theories; in an article
in Phil. Mag. 29 (1915) he offers a theory involving an attempt on supra-conduc-
tivity, which the others do not touch.

The field of thcrmionics is thoroughly covered in Richardson's Emission of Elec-
tricity from Hot Bodies (2d edition, 1921). Subsequent theoretical papers by Rich-
ardson are in Proc. Roy. Soc. A 105, pp. 387-405 (1924) and Proc. Phys. Soc, London,
86, pp. 383-399 (1924), and one by H. A. Wilson on what I have called the "thermo-
dynamical argument" in Phys. Rev. (2) 24, pp. 38-48 (1924). For various interpre-
tations of the surface double-layers see Debye, Ann. d. Phys. 33, pp. 440-489 (1910);
Schottky, ZS. f. Phys. U, pp. 63-106 (1923); and Frenkcl, Phil. Mag. 33, pp.297-322
(1917). Germer's investigation of the distribution-in-energy of thermionic electrons
is briefly reported in Science, 4$, 392 (1923), and a fuller account is to be published;
Davisson and Germer's determination of L in Phys. Rev. 20, pp. 300-330 (1922).
For the photoelectric measurements establishing equation (12), consult Millikan,
Phys. Rev. 7, pp. 355-388 (1916); for the relation between values of P and contact-
potential-difference consult Page, Am. Journ. Sci. 3(1, pp. 501—508 (1913) and Milli-
kan, Phys. Rev. 7, pp. 18-32 ( 1916). Values of the thermionic constant b are tabu-
lated in Richardson's book and in Dushman's article, Phys. Rev. (2) 21, pp. 623-636
(1923). Values of the photoelectric constant P are tabulated by Kirchner, Phys.
ZS. 25, pp. 303-306 (1924) and by Hamer (Journ. Opt. Soc. 9, pp. 251-257 (1924).
For the arguments that the constant C of equation (23) is*a universal constant,
consult the references given by Dushman (/. c. supra) and Richardson, Phys. Rev.
(2) 23, pp. 153-155 (1924); for the data, Dushman in Phys. Rev. (2) 23, p. 156 (1924).